Representation theory | Representation theory of groups | Group theory
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system. The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the automorphism group of an object. If the object is a vector space we have a linear representation. Some people use realization for the general notion and reserve the term representation for the special case of linear representations. The bulk of this article describes linear representation theory; see the last section for generalizations. (Wikipedia).
Representation theory: Introduction
This lecture is an introduction to representation theory of finite groups. We define linear and permutation representations, and give some examples for the icosahedral group. We then discuss the problem of writing a representation as a sum of smaller ones, which leads to the concept of irr
From playlist Representation theory
Visual Group Theory, Lecture 1.4: Group presentations
Visual Group Theory, Lecture 1.4: Group presentations We begin this lecture by learning how to take a Cayley diagram and label its nodes with the elements of a group. Such a labeled diagram can function as a "group calculator". It leads to the notion of a "group presentation", which is a
From playlist Visual Group Theory
Definition of a group Lesson 24
In this video we take our first look at the definition of a group. It is basically a set of elements and the operation defined on them. If this set of elements and the operation defined on them obey the properties of closure and associativity, and if one of the elements is the identity el
From playlist Abstract algebra
RT6. Representations on Function Spaces
Representation Theory: We note how to transfer a group action of a group G on a set X to a group action on F(X), the functions on X. Because F(X) is a vector space, we obtain a representation of the group, and we can apply previous techniques. In particular, the group acts on itself in
From playlist Representation Theory
Representations of Finite Groups | Definitions and simple examples.
We define the notion of a representation of a group on a finite dimensional complex vector space. We also explore one and two dimensional representations of the cyclic group Zn. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Personal Website: http://www.mich
From playlist Representations of Finite Groups
Visual Group Theory, Lecture 1.6: The formal definition of a group
Visual Group Theory, Lecture 1.6: The formal definition of a group At last, after five lectures of building up our intuition of groups and numerous examples, we are ready to present the formal definition of a group. We conclude by proving several basic properties that are not built into t
From playlist Visual Group Theory
RT1: Representation Theory Basics
Representation Theory: We present basic concepts about the representation theory of finite groups. Representations are defined, as are notions of invariant subspace, irreducibility and full reducibility. For a general finite group G, we suggest an analogue to the finite abelian case, whe
From playlist *** The Good Stuff ***
Group actions in abstract algebra
In this first video on group actions, I use an example of some previous work on the symmetric group to give you some intuition about group actions. Beware when reading your textbook. It is probably unnecessary difficult just due to the dot notation that is used when describing group acti
From playlist Abstract algebra
Visual Group Theory, Lecture 5.1: Groups acting on sets
Visual Group Theory, Lecture 5.1: Groups acting on sets When we first learned about groups as collections of actions, there was a subtle but important difference between actions and configurations. This is the tip of the iceberg of a more general and powerful concept of a group action. Ma
From playlist Visual Group Theory
Representations of p-adic reductive groups by Tasho Kaletha
PROGRAM ZARISKI-DENSE SUBGROUPS AND NUMBER-THEORETIC TECHNIQUES IN LIE GROUPS AND GEOMETRY (ONLINE) ORGANIZERS: Gopal Prasad, Andrei Rapinchuk, B. Sury and Aleksy Tralle DATE: 30 July 2020 VENUE: Online Unfortunately, the program was cancelled due to the COVID-19 situation but it will
From playlist Zariski-dense Subgroups and Number-theoretic Techniques in Lie Groups and Geometry (Online)
Lie Groups and Lie Algebras: Lesson 41: Elementary Representation Theory I
Lie Groups and Lie Algebras: Lesson 41: Elementary Representation Theory I I wanted to begin a more intricate example of the principle of a Universal Covering group, but I think I need to cover a little background material. We need to get a grip on what is meant by "Representation Theory"
From playlist Lie Groups and Lie Algebras
Representations of finite groups of Lie type (Lecture 1) by Dipendra Prasad
PROGRAM : GROUP ALGEBRAS, REPRESENTATIONS AND COMPUTATION ORGANIZERS: Gurmeet Kaur Bakshi, Manoj Kumar and Pooja Singla DATE: 14 October 2019 to 23 October 2019 VENUE: Ramanujan Lecture Hall, ICTS Bangalore Determining explicit algebraic structures of semisimple group algebras is a fun
From playlist Group Algebras, Representations And Computation
Representation Theory(Repn Th) 3 by Gerhard Hiss
DATE & TIME 05 November 2016 to 14 November 2016 VENUE Ramanujan Lecture Hall, ICTS Bangalore Computational techniques are of great help in dealing with substantial, otherwise intractable examples, possibly leading to further structural insights and the detection of patterns in many abstra
From playlist Group Theory and Computational Methods
Henniart: Classification des représentations admissibles irréductibles modulo p...
Recording during the thematicmeeting : "Algebraic and Finite Groups, Geometry and Representations. Celebrating 50 Years of the Chevalley Seminar " the September 23, 2014 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this
From playlist Partial Differential Equations
Geordie Williamson 6 August 2020
Topic: Modular Representation Theory and Geometry Abstract: This will be a broad survey talk on interactions between geometry and representation theory, with a focus on representations in positive characteristic (“modular representation theory”). I will outline several basic questions (e.
From playlist Geordie Williamson external seminars
Parahoric Subgroups and Supercuspidal Representations of p-Adic groups - Dick Gross
Dick Gross Harvard University December 9, 2010 This is a report on some joint work with Mark Reeder and Jiu-Kang Yu. I will review the theory of parahoric subgroups and consider the induced representation of a one-dimensional character of the pro-unipotent radical. A surprising fact is th
From playlist Mathematics
On characterization of monomial irreducible representations by Pooja Singla
DATE & TIME 05 November 2016 to 14 November 2016 VENUE Ramanujan Lecture Hall, ICTS Bangalore Computational techniques are of great help in dealing with substantial, otherwise intractable examples, possibly leading to further structural insights and the detection of patterns in many abstra
From playlist Group Theory and Computational Methods
This video contains the origins of group theory, the formal definition, and theoretical and real-world examples for those beginning in group theory or wanting a refresher :)
From playlist Summer of Math Exposition Youtube Videos
CTNT 2022 - An Introduction to Galois Representations (Lecture 3) - by Alvaro Lozano-Robledo
This video is part of a mini-course on "An Introduction to Galois Representations" that was taught during CTNT 2022, the Connecticut Summer School and Conference in Number Theory. More about CTNT: https://ctnt-summer.math.uconn.edu/
From playlist CTNT 2022 - An Introduction to Galois Representations (by Alvaro Lozano-Robledo)